The standard twist of $L$-functions revisited
The analytic properties of the standard twist $F(s,\alpha )$, where $F(s)$ belongs to a wide class of $L$-functions, are of prime importance in describing the structure of the Selberg class. In this paper we present a deeper study of such properties. In particular, we show that $F(s,\alpha )$ satisfies a functional equation of a new type, somewhat resembling that of the Hurwitz–Lerch zeta function. Moreover, we detect the finer polar structure of $F(s,\alpha )$, characterizing in two different ways the occurrence of finitely or infinitely many poles as well as giving a formula for their residues.